12 



work required to move a unit of mass against the force, from P to C. 

 If the mass be moved along the radius PC, the radial component is 

 the only part of the force against which work is done. The radial 



O 



Figure 4. 



component of the lunar tide-producing force is as shown in equation 



(5): 



MM(r/53)(3cos2 0-l) 



As derived, this force is positive in the direction CP. 

 The lunar tide-producing potential at P is therefore: 



Vt=- rMM(r/i^')(3 cos2 e-l)dr^-(Mfji/R^){3 cos^ e-l)jjdr 

 = JiM^fxiryK') (3 cos^ ^~ 1) (1 1) 



23. Relation of potential to jorce. — It follows from the definition of 

 the potential of a force, that its rate of change, in any direction, is 

 the component of the force acting in that direction. Thus the rate 

 of change of the lunar tide-producing potential in a direction per- 

 pendicular to the radius (in the plane of the moon, the point, and 

 the earth's center) is: 



dVtld{:rd)=dVtlrde=)iMu.{rlR^)d{2, cos^ 6-1) /dd 



=^-3Mfx(r/R^) cos 6 sin 6= -3/2 Mii(,rlR^) sin 26 



as found in equation (6). The negative sign results from the fact 

 that the direction of the force is opposite to the direction in which 

 6 is increasing, as will be apparent from a reference to figure 1. 



24. It is evident from the preceding paragraph that when the po- 

 tential varies from point to point over a water surface, such as the 

 surface of the oceans, the water tends to move from areas of low 

 potential toward areas of high potential, just as it would tend to 

 move from areas having a higher elevation toward the areas having a 

 lower elevation. When a water surface is in equilibrium, the total 

 potential of all forces acting upon it evidently must be the same at 

 all points on the surface. 



25. The lunar tide-producing potential at any point P on the sur- 

 face of the earth is found at once by substituting the earth's radius 

 a for r m equation (11), and is: 



Fi= KMM(a7E') (3 cos^ 6- 1) (12) 



